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7. x = y + 2z - w = -1 2x + y2z - 2w = -2 -x + 2y4z + w = 1 3x - 3w = -3

7. x = y + 2z - w = -1 2x + y2z - 2w = -2 -x + 2y4z + w = 1 3x - 3w = -3

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Instruction: Solving Linear Systems by Gaussian Elimination** **Instructions:** In Exercises 5–8, solve the system by Gaussian elimination. For detailed steps on how to perform Gaussian elimination, refer to the associated tutorial links and example problems provided on the site. Gaussian elimination involves transforming a system of linear equations into a row-echelon form using elementary row operations, and then performing back-substitution to find the solution.
Transcribed Image Text:**Instruction: Solving Linear Systems by Gaussian Elimination** **Instructions:** In Exercises 5–8, solve the system by Gaussian elimination. For detailed steps on how to perform Gaussian elimination, refer to the associated tutorial links and example problems provided on the site. Gaussian elimination involves transforming a system of linear equations into a row-echelon form using elementary row operations, and then performing back-substitution to find the solution.
### System of Linear Equations The given system of linear equations can be written in matrix or standard form for the purpose of analysis or solving. Here, we have a set of four equations with four unknowns: \( x \), \( y \), \( z \), and \( w \). The equations are as follows: 1. \( x - y + 2z - w = -1 \) 2. \( 2x + y - 2z - 2w = -2 \) 3. \( -x + 2y - 4z + w = 1 \) 4. \( 3x - 3w = -3 \) ### Explanation Each equation represents a linear combination of the variables \( x \), \( y \), \( z \), and \( w \). - In the first equation, the coefficients of \( x \), \( y \), \( z \), and \( w \) are 1, -1, 2, and -1, respectively. - In the second equation, the coefficients are 2, 1, -2, and -2. - In the third equation, the coefficients are -1, 2, -4, and 1. - In the fourth equation, the coefficients of \( x \) is 3, the coefficients of \( y \) and \( z \) are 0 (implying these variables are absent), and the coefficient of \( w \) is -3. This type of problem typically would be approached by solving the system using methods such as: 1. **Substitution**: Solving one of the equations for one variable and then substituting that expression into the other equations. 2. **Elimination**: Adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variables. 3. **Matrix Methods**: Representing the equations as matrices and using techniques like Gaussian elimination or finding the inverse matrix (if it exists). ### Practical Application Systems of linear equations are widely used in various fields including engineering, physics, computer science, and economics to model and solve problems regarding rates, balances, and dynamic systems. Solving these systems provides insights into the relationships between different variables and can often help in optimizing performance within given constraints. Feel free to experiment with different techniques to find the solutions for \( x \), \( y \), \( z \), and \( w \
Transcribed Image Text:### System of Linear Equations The given system of linear equations can be written in matrix or standard form for the purpose of analysis or solving. Here, we have a set of four equations with four unknowns: \( x \), \( y \), \( z \), and \( w \). The equations are as follows: 1. \( x - y + 2z - w = -1 \) 2. \( 2x + y - 2z - 2w = -2 \) 3. \( -x + 2y - 4z + w = 1 \) 4. \( 3x - 3w = -3 \) ### Explanation Each equation represents a linear combination of the variables \( x \), \( y \), \( z \), and \( w \). - In the first equation, the coefficients of \( x \), \( y \), \( z \), and \( w \) are 1, -1, 2, and -1, respectively. - In the second equation, the coefficients are 2, 1, -2, and -2. - In the third equation, the coefficients are -1, 2, -4, and 1. - In the fourth equation, the coefficients of \( x \) is 3, the coefficients of \( y \) and \( z \) are 0 (implying these variables are absent), and the coefficient of \( w \) is -3. This type of problem typically would be approached by solving the system using methods such as: 1. **Substitution**: Solving one of the equations for one variable and then substituting that expression into the other equations. 2. **Elimination**: Adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variables. 3. **Matrix Methods**: Representing the equations as matrices and using techniques like Gaussian elimination or finding the inverse matrix (if it exists). ### Practical Application Systems of linear equations are widely used in various fields including engineering, physics, computer science, and economics to model and solve problems regarding rates, balances, and dynamic systems. Solving these systems provides insights into the relationships between different variables and can often help in optimizing performance within given constraints. Feel free to experiment with different techniques to find the solutions for \( x \), \( y \), \( z \), and \( w \
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